3.7.2 The Complex Index of Refraction

In looking in detail at the polarization of dielectrics, we switched from a simple dielectric contant e_r to a dielectric function e_r(w) = e' + ie''. This, after some getting used to, makes life much easier and provides for new insights not easily obtainable otherwise.

We now do exactly the same thing for the index of refraction, i.e. we replace n by a complex index of refraction n*.

n* = n + i · k

We use the old symbol n for the real part instead of n' and k instead of n'', but that is simply to keep with tradition.

With the dielectric constant and a constant index of refraction we had the basic relation ,

n² = e_r

We simply use this relation now for defining the complex index of refraction. This gives us

(n + ik)²	=	e' + i · e''

With n = n(w); k = k(w), since e' and e'' are frequency dependent as discussed before.

Re-arranging for n and k yields somewhat unwieldy equations:

n²	=	1 2	æ ç è	æ è	e' ² + e'' ²	ö ø	½	+ e'	ö ÷ ø

k²	=	1 2	æ ç è	æ è	e' ² + e'' ²	ö ø	½	– e'	ö ÷ ø

Anyway - That is all. We now have optics covered. An example of an real complex index of refraction is shown in the link.

So lets see how it works and what k, the so far unspecified imaginary part of n_com, will give us.

First, lets get some easier formula. In order to do this, we remember that e'' was connected to the conductivity of the material and express e'' in terms of the (total) conductivity as

e''	=	s_DK e₀ · w

Note that in contrast to the definition of e'' given before in the context of the dielectric function, we have an e₀ in the e'' part. We had, for the sake of simplicity, made a convention that the e in the dielectric function contain the e₀, but here it more convenient to write it out, because then e' = e₀ · e_r is reduced to e_r and directly related to the "simple" index of refraction n

Using that in the expression (n + ik)² gives

(n + ik)²	=	n² – k² + i · 2nk	= e' + i ·	s_DK e₀ · w

We have a complex number on both sides of the equality sign, and this demands that the real and imaginary parts must be the same on both sides, i.e.

n² – k²	=	e'

nk	=	s_DK 2e₀w

Separating n and k finally gives

n²	=	1 2	æ ç è	e'	+	æ è	e' ² +	s_DK² 4e₀²w²	ö ø	½	ö ÷ ø

k²	=	1 2	æ ç è	– e'	+	æ è	e' ² +	s_DK² 4e₀²w²	ö ø	½	ö ÷ ø

Similar to what we had above, but now with basic quantities like the "dielectric constant" e' = e_r and the conductivity s_DK².

The equations above go beyond just describing the optical properties of (perfect) dielectrics because we can include all kinds of conduction mechanisms into s, and all kinds of polarization mechanisms into e'.

We can even use these equations for things like the reflectivity of metals, as we shall see.

Keeping in mind that typical n's in the visible region are somewhere between 1.5 - 2.5 (n » 2.5 for diamond is one of the higher values as your girl friend knows), we can draw a few quick conclusions: From the simple but coupled equations for n and k follows:

k should be rather small for "common" optical materials, otherwise our old relation of n = (e_r)^½ would be not good.

k should be rather small for "common" optical materials, because optical materials are commonly insulators, i.e. s_DK » 0 applies.

For s_DK = 0 (and, as we would assume as a matter of course, e_r > 0) we obtain immediately n = (e_r)^½ and k = 0 - the old-fashioned simple relation between just e_r and n.

For large s_DK values, both n and k will become large. We don't know yet what k means in physical terms, but very large n simply mean that the intensity of the reflected beam approaches 100 %. Light that hits a good conductor thus will get reflected - well, that is exactly what happens between light and (polished) metals, as we know from everyday experience.

But now we must look at some problems that can be solved with the complex index of refraction in order to understand what it encodes.